#include <stdio.h>
#include <math.h>

#define T 4.0
double f(double y){
    return y*cos(y);
}
double df(double y){
    return cos(y)-y*sin(y);
}
double taylor2(int N,double y){
    double h=T/N;
    for(int k=1;k<=N;k++){
        y=y+f(y)*h+df(y)*f(y)*h*h/2;
    }
    return y;
}
double euler(int N,double y){
    double h=T/N;
    for(int k=1;k<=N;k++){
        y=y+f(y)*h;
    }
    return y;
}
int main(){
    double yold=1;
    for(int n=4;n<=1024*1024;n=n*2){
        double y=taylor2(n,1.0);
        printf("n=%d yn=%g yn-yold=%.14e\n",
            n,y,y-yold);
        yold=y;
    }
    return 0;
}